📈 Capital Asset Pricing Model (CAPM)

What CAPM is and what it is used for
Who developed it, and the Black (zero-beta) version
The SML and the CAPM formula
Using CAPM for asset pricing and valuation
Beta and the asset-specific required return
Systematic vs. unsystematic risk and diversification
The efficient frontier
Assumptions of the model
Criticisms, empirical evidence, modified betas and extensions

💡 Core Idea

In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio.

The model takes into account three things:

The asset's sensitivity to non-diversifiable risk (also known as systematic risk or market risk), often represented by the quantity beta ( $\beta$ ) in the financial industry.
The expected return of the market.
The expected return of a theoretical risk-free asset.

Conditions Under Which CAPM Works

CAPM assumes one of the following (plus zero transaction costs):

A particular form of utility function in which only the first and second moments matter, so risk is measured by variance (for example, a quadratic utility).
Or, alternatively, asset returns whose probability distributions are completely described by the first two moments (for example, the normal distribution).
Zero transaction costs, which are necessary for diversification to get rid of all idiosyncratic risk.

Under these conditions, CAPM shows that the cost of equity capital is determined only by beta.

Why It Is Still Used

It has failed numerous empirical tests.
More modern approaches to asset pricing and portfolio selection exist, such as arbitrage pricing theory and Merton's portfolio problem.
Even so, CAPM remains popular because of its simplicity and utility in a variety of situations.

🧑‍🔬 Inventors and History

CAPM was introduced independently by economists:
- Jack Treynor (1961, 1962)
- William F. Sharpe (1964)
- John Lintner (1965)
- Jan Mossin (1966)
They built on the earlier work of Harry Markowitz on diversification and modern portfolio theory.
Sharpe, Markowitz and Merton Miller jointly received the 1990 Nobel Memorial Prize in Economic Sciences for this contribution to financial economics.

Black CAPM (Zero-Beta CAPM)

Fischer Black (1972) developed another version of CAPM, called Black CAPM or zero-beta CAPM.
It does not assume the existence of a riskless asset.
It replaces the risk-free rate with the return of a "zero-beta portfolio", a portfolio that has no correlation with the market.
It was found to be more robust against empirical testing, particularly in explaining why the security market line is often flatter than the standard model predicts.
It was influential in the widespread adoption of the CAPM.

📐 The Formula and the Security Market Line (SML)

The CAPM is a model for pricing an individual security or portfolio. For individual securities, it uses the security market line (SML) and its relation to expected return and systematic risk (beta) to show how the market must price individual securities in relation to their security risk class.

The CAPM / SML Equation

E(R_i) = R_f + \beta_i \left( E(R_M) - R_f \right)

Symbol	Meaning
$E(R_i)$	Expected (required) return on asset $i$
$R_f$	Risk-free rate (the intercept of the SML)
$\beta_i$	Sensitivity of asset $i$ to non-diversifiable market risk
$E(R_M)$	Expected return of the market
$E(R_M) - R_f$	Market risk premium (the slope of the SML)

Reward-to-Risk Idea

The SML enables us to calculate the reward-to-risk ratio for any security in relation to that of the overall market.
When the expected rate of return for any security is deflated by its beta coefficient, the reward-to-risk ratio for any individual security in the market equals the market reward-to-risk ratio.

Reading the SML

The SML graphs the results from the CAPM formula.
- x-axis: risk (beta)
- y-axis: expected return
The intercept is the nominal risk-free rate available for the market.
The slope is the market premium, $E(R_m) - R_f$ . It illustrates the trade-off between risk and return.
The SML can be regarded as a single-factor model of the asset price, where $\beta$ is the exposure to changes in the value of the Market.
In equilibrium, all assets should plot directly on the SML.

Position of the security	Interpretation	Why
Above the SML	Undervalued	The investor can expect a greater return for the inherent risk
On the SML	Fairly priced (equilibrium)	Return matches the systematic risk
Below the SML	Overvalued	The investor would be accepting less return for the amount of risk assumed

By determining the position of a security relative to this line, investors can identify whether the expected return justifies the asset's market-related volatility.
It is a useful tool for determining whether an asset being considered for a portfolio offers a reasonable expected return for its risk.

💰 Asset Pricing with CAPM

Comparing Required vs. Estimated Return

Once the expected/required rate of return $E(R_i)$ is calculated using CAPM, it can be compared to the asset's estimated rate of return over a specific investment horizon to decide whether it is an appropriate investment.
This comparison needs an independent estimate of the return outlook for the security, based on fundamental or technical analysis techniques, including P/E, M/B and so on.

Correct Pricing

Assuming that the CAPM is correct:

An asset is correctly priced when its estimated price is the same as the present value of future cash flows of the asset, discounted at the rate suggested by CAPM.
If the estimated price is higher than the CAPM valuation, the asset is overvalued.
If the estimated price is below the CAPM valuation, the asset is undervalued.
If the asset does not lie on the SML, this could also suggest mis-pricing.

Expected Return at Time $t$

E(R_t) = \frac{E(P_{t+1}) - P_t}{P_t}

A higher expected return than what CAPM suggests indicates that $P_t$ is too low (the asset is currently undervalued).
This assumes that at time $t+1$ the asset returns to the CAPM-suggested price.

Certainty Equivalent Pricing Formula

The asset price $P_0$ using CAPM, sometimes called the certainty equivalent pricing formula, is a linear relationship given by:

P_0 = \frac{1}{1+R_f}\left[ E(P_T) - \frac{\mathrm{Cov}(P_T, R_M)\,\left(E(R_M) - R_f\right)}{\mathrm{Var}(R_M)} \right]

where $P_T$ is the future price of the asset or portfolio.

🎚️ Asset-Specific Required Return and Beta

The CAPM returns the asset-appropriate required return or discount rate: the rate at which future cash flows produced by the asset should be discounted, given that asset's relative riskiness.

Beta	Riskiness	Discount rate
Greater than 1	More than average "riskiness"	Higher
Less than 1	Lower than average risk	Lower
Equal to 1	Same as the market	Market-level

A more risky stock has a higher beta and is discounted at a higher rate. Less sensitive stocks have lower betas and are discounted at a lower rate.
Given the accepted concave utility function, CAPM is consistent with the intuition that investors require a higher return for holding a more risky asset.

The Market's Beta

Since beta reflects asset-specific sensitivity to non-diversifiable market risk, the market as a whole, by definition, has a beta of one.
Stock market indices are frequently used as local proxies for the market, and in that case, by definition, have a beta of one.
An investor in a large, diversified portfolio, such as a mutual fund designed to track the total market, therefore expects performance in line with the market.

Fundamental Valuation (Present Value)

Once the required expected return $E(R_i)$ is established via CAPM, it is used as the discount rate to determine an asset's intrinsic value based on future cash flows (CF).

PV = \sum_{t=1}^{n} \frac{E(CF_t)}{\left(1 + E(R_i)\right)^t}

An asset is undervalued if its calculated present value is higher than the current market price.
An asset is overvalued if the price exceeds this intrinsic value.

⚖️ Risk and Diversification

Two Kinds of Risk

Type	Also called	Description
Systematic risk	Undiversifiable risk, market risk	Risk common to all securities
Unsystematic risk	Idiosyncratic risk, diversifiable risk	Risk associated with individual assets

The risk of a portfolio comprises both systematic and unsystematic risk.
Unsystematic risk can be diversified away to smaller levels by including a greater number of assets in the portfolio, as specific risks "average out".
The same is not possible for systematic risk within one market.

How Many Securities Are Enough?

In developed markets such as the UK or US, a portfolio of approximately 30-40 securities renders the portfolio sufficiently diversified so that risk exposure is limited to systematic risk only.
The number may vary depending on how securities are weighted, which alters the overall risk contribution of each security.
- Example: market cap weighting means securities of companies with larger market capitalization take up a larger portion of the portfolio, making it effectively less diversified.
In developing markets, a larger number of securities is required for diversification because of higher asset volatilities.

What Is Rewarded

A rational investor should not take on any diversifiable risk, as only non-diversifiable risks are rewarded within the scope of this model.
The required return on an asset must be linked to its contribution to overall portfolio riskiness, as opposed to its "stand alone risk".
In the CAPM context, portfolio risk is represented by higher variance, that is, less predictability.
The beta of the portfolio is the defining factor in rewarding the systematic exposure taken by an investor.

🏔️ Efficient Frontier

CAPM assumes that the risk-return profile of a portfolio can be optimized. An optimal portfolio displays the lowest possible level of risk for its level of return.
Since each additional asset introduced into a portfolio further diversifies it, the optimal portfolio must comprise every asset, with each asset value-weighted to achieve efficiency.
- This assumes no trading costs and that any asset is infinitely divisible.
All such optimal portfolios, one for each level of return, make up the efficient frontier.
Because unsystematic risk is diversifiable, the total risk of a portfolio can be viewed as beta.

📋 Assumptions

All investors:

Aim to maximize economic utilities (asset quantities are given and fixed).
Are rational and risk-averse.
Are broadly diversified across a range of investments.
Are price takers, that is, they cannot influence prices.
Can lend and borrow unlimited amounts under the risk-free rate of interest.
Trade without transaction or taxation costs.
Deal with securities that are all highly divisible into small parcels (all assets are perfectly divisible and liquid).
Have homogeneous expectations.
Have all information available at the same time.

🚧 Criticisms and Empirical Evidence

In their 2004 review, economists Eugene Fama and Kenneth French argue that:

"the failure of the CAPM in empirical tests implies that most applications of the model are invalid".

Despite its theoretical importance, the CAPM is often criticized for failing to match real-world market dynamics.
The criticisms cover empirical failures, theoretical inconsistencies and behavioral critiques.

Modified Betas

There has been research into a mean-reverting beta, often referred to as the adjusted beta, as well as the consumption beta.
In empirical tests, the traditional CAPM did as well as or outperformed these modified beta models.
Mankiw and Shapiro (1986) found that the market beta of the traditional CAPM outperformed the consumption beta in explaining the cross-section of stock returns.
The adjusted beta (Blume's method) improves forecast accuracy, but it does not consistently resolve the underlying empirical failures of the model's predictive power over long horizons.

🔧 Extensions of the CAPM

Several extensions have been developed to address the empirical and theoretical limitations of the standard CAPM by relaxing the model's core assumptions.
Example: investors with longer-term outlooks might optimally choose long-term inflation-linked bonds instead of short-term rates. This suggests that the relevant risk-free rate depends on the investor's horizon.

🧠 Quick Review

Concept	Key point
CAPM	Gives a theoretically appropriate required return for an asset
Beta	Measures sensitivity to non-diversifiable (market) risk
SML	Plots expected return against beta; slope is the market risk premium
Above / below SML	Undervalued / overvalued
Black CAPM	Replaces the risk-free rate with a zero-beta portfolio return

Capital Asset Pricing Model: Finance Study Notes